Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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33 views

Covariance of $X$ and $Y^2$ where $(X,Y)$ is bivariate normal

I'm trying to solve a case where there is bivariate random vector $(X,Y)$ that has the bivariate normal distribution below ($-1<\rho<1$): $$\begin{pmatrix} X\\ Y \end{pmatrix}\sim N_{2}\...
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22 views

find the correlation coefficient of two random variables after the same function transformation

we have two random variables: U1 and U2 follow uniform distribution between 0 and 1: U1 ~ U(0,1), U2 ~ U(0,1) and correlation: corr(U1,U2) = ρ covariance : cov(U1,U2) = corr(U1,U2)/12 Then we do ...
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25 views

Bivariate normal distribution aX+bY [closed]

$(X, Y) \sim$ Bivariate Normal I want to find out probability function of $aX+bY$ without using the moment generating function.
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32 views

Finding the Joint Distribution of Two Normally Distributed Random Variables

Question: Suppose $X_1$, $X_2$ and $X_3$ are independent random variables such that $X_1 \sim N(0,1)$, $X_2 \sim N(1,4)$ and $X_3 \sim N(-1,2)$. Let $Y_1=X_1+X_3$ and $Y_2=X_1+X_2-2X_3$. Give the ...
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1answer
33 views

Multivariate normal distribution not uniquely determined by means and variance?

Suppose $X_1,X_2$ are independent standard normal distributions, and suppose $\alpha = (X_1, X_1 \cos\theta + X_2 \sin\theta), \beta = ( X_1 \cos\theta + X_2 \sin\theta, X_1)$. Then both $\alpha$ and ...
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8 views

convolution between normal bivariate and gaussian

i need to perform the convolution between a normal bivariate and a gaussian. Both are normalized. I expect that the result will be a normal bivariate, right ? Which would be the variances of that new ...
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10 views

conditional distribution of convolution of two normal bivariate

i have two normal bivariate f(x,y) and g(x,y). I want to calculate their convolution h=f*g and i'm interested in the conditional distribution of the convoluted bivariates. How is related the ...
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12 views

product of normal bivariate

Suppose f(x,y) and g(x,y) are bivariate normal. You are given E(X), E(Y), Var(X) and Var(Y) for both f(x,y) and g(x,y). Is there a simple way to estimate E(X), E(Y), Var(X) and Var(Y) of the product f(...
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1answer
15 views

Sampling from a Bivariate Cauchy?

Given the bivariate Cauchy distribution: $$f(x,y; x_0, y_0) \sim \frac{1}{2\pi}\frac{1}{((x - x_0)^2 + (y - y_0)^2+1)^{1.5}}$$ How do you generate samples appropriately? I am aware of inverse ...
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73 views

What is the probability $P(X>0\mid X+Y>0)$ when $X$ and $Y$ are both independent normal mean zero with different variances? [duplicate]

Let $X$ have standard deviation $\sigma$ and $Y$ standard deviation $\tau$. I can write $$P(X>0\mid X+Y>0) = \frac{P(X>0, X+Y>0)}{P(X+Y>0)}$$ As all variables are zero-mean and ...
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1answer
7 views

does marginalization of bivariate normal distribution always yield a normal distribution?

Let's Suppose (X,Y) are bivariate random variables with a bivariate normal distribution, and it has a pdf of: $$f_{XY}(x,y) = \frac{1}{2\pi\sigma_x \sigma_y (1-\rho^2)^{1/2}}\text{exp}\bigg(-\frac{1}{...
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1answer
16 views

showing that a r.v. is a bivariate normal distribution…

A bullet is fired at a target. (X,Y) is a bivariate normal random variable that represent the horizontal and vertical distance from the center of the target where the bullet strikes the target when ...
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1answer
13 views

Joint distribution of sum of random variables in bivariate distribution

We have the following bivariate distribution: $$f(x,y) = e^{-(\theta x + y/\theta)}$$ for x,y >0. We would like to find the distribution of $$\left(\sum_{i=1}^n X_i, \sum_{i=1}^n Y_i\right)$$ I'm not ...
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1answer
30 views

Finding $P(X+Y \geq 0.5)$ given $f(x,y) = x - y + 1$ for $0 \leq x, y \leq 1$

For the bivariate density function (not necessarily independent) $$f(x,y) = x - y + 1$$ for $$0 \leq x, y \leq 1$$ I am trying to find $\Pr(X+Y \geq 0.5)$. I integrated $y$ across $0.5-x$ to $1$ ...
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2answers
134 views

Bivariate moment generating function

For two random variables $(X,Y)$, the MGF can be defined as $M_{XY}(s,t) = E[e^{sX+tY}]$. Find $M_{XY}(s,t)$ when $X$ and $Y$ are two jointly normal random variables with $E[X] = μ_X,E[Y] = μ_Y ,Var(...
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1answer
22 views

Intuitively, *why* is the joint density function zero when the joint cumulative function is constant/single variable?

Suppose I have a CDF of the form: $$F_{XY}(x,y)=\begin{cases} g(x,y) & x\le \frac {1}{2} , \ y\le \frac {1}{2}\mathstrut \mathstrut \\[5pt] g(x, \frac 12\mathstrut ) & x\le \frac 12, \ y &...
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2answers
50 views

Finding the joint distribution of two dependent random variables given conditions

Suppose $X \sim U(0,1)$ and $Y|X=x \sim U(0,1-x)$. Find the joint distribution of $(X,Y)$ given that $X\le \frac{1}{2}$, $Y\le \frac{1}{2}.$ My attempt: The pdf of $X$ is $f_{X}(x) = 1$ for $x \...
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1answer
27 views

Name this distribution

The bi-variate Cauchy distribution is given by $$ \mathbb{R}^2\ni(x,y)\mapsto\frac{1}{2\pi}\frac{1}{(x^2+y^2+1)^{1.5}}\,. $$ What is the name of the following probability distribution: $$ \mathbb{R}^...
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56 views

Finding Joint PMF of bivariate disttibution

Problem: Let $W$ equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose $P(W<1)=0.02$ and $P(W>1.072)=0.08$. Call a box of soap light, good, or heavy ...
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34 views

Question about obtaining a bivariate normal distribution

A random vector $X = (X_1 , X_2 ) \in R^2$ is distributed according to the bivariate normale distribution with mean vector $\mu_x = (0,0)$ and a covariance matrix $$\Sigma_x = \begin{bmatrix} 1 &...
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2answers
80 views

Determine whether $f(x,y)=\sqrt{|xy|}$ and/or $g(x,y)=e^{|x|^3y}$ are differentiable at the point $(0, 0)$.

Determine whether $$f(x,y)=\sqrt{|xy|}$$ and/or $$g(x,y)=e^{|x|^3y}$$ are differentiable at the point $(0, 0)$. Also, find its total derivative if it exists at (0, 0); if not, prove that it is not ...
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26 views

Conditional Distribution inference

Let $X, Y$ have a joint uniform distribution on the unit square. Thus, $$f_{X\mid Y}(x\mid y) = \begin{cases} 1 &0\leq x\leq1 \\ 0 & \text{otherwise} \end{cases}$$ ...
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75 views

Estimating two means of bivariate gaussian

Consider a bivariate gaussian distribution, with parameters $\mu_1$ and $\mu_2$ for the two unknown means, and $\sigma_1$, $\sigma_2$ and $\rho$ for the known covariance matrix, \begin{align} \...
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25 views

Sum of correlated log-normal distributions

Suppose $P(x_1,x_2)$ is the pdf of a bivariate normal distribution, where $x_1\sim\mathcal{N}(\mu_1,\sigma_1^2)$ and $x_2\sim\mathcal{N}(\mu_2,\sigma_2^2)$ and the correlation between $x_1$ and $x_2$ ...
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42 views

Example of R$^2$-valued r.v. with each component being univariate normal, but not being itself bivariate normal

I quote Jacod-Protter referring to an example of an $\mathbb{R}^2$-valued r.v. $X = (Y, Z)$, with each component being univariate, but not being itself multivariate normal.$ Let $\mathcal{L}$(Y) ...
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1answer
43 views

Bivariate Probability Question - Not sure where to start

Two points are selected randomly on a line of length 30 so as to be on opposite sides of the midpoint line. In other words, the two points X and Y are independent random variables such that X is ...
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1answer
27 views

Finding Marginal PDFs of $X$ and $Y$ from the joint pdf $f(x,y)$

I am studying for a probability exam and looking over an old question that goes as follows: Let $f(x,y)=2e^{-x-y}, 0\le x\le y < \infty$. From here, I understand that the marginal PDFs of $X$ and ...
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7 views

Generalized equation for bivariate normal distribution

I would like to know how to derive the CDF of a bivariate probit model as shown below: $p^{y_{1},y_{2}} = \Phi(2c_{1} - 1*x_{1}\beta_{1},2c_{2} - 1*x_{2}\beta_{2}, (2c_{1} - 1)*(2c_{2} - 1)*\rho)$ ...
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0answers
15 views

Expectation of Indicatior with Bivariate Normal

Let (X,Y) be bivariate normal with correlation $\rho$. I am asked to calculate: \begin{align*} \mathbb{E}[e^X \mathbb{I}_{X\leq Y}] \end{align*} I have tried plugging in the pdf but it's a mess. any ...
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1answer
44 views

How to find joint CDF of uniform bivariate density over triangle?

I Have that (X,Y) are random variables that has uniform density over $\Omega=((x,y):x\geq 0, y\geq0, x+y\leq 1) $. Using the steps given in this related question: Joint PDF of two random variables ...
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1answer
108 views

The distribution of the sum of three random variables

Consider three rv, X, Y, Z with joint Gaussian distribution. Namely, the first two are a bivariate Gaussian vector, with non zero correlation ρ(x,y) , while Z is independent from the first two. ...
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22 views

Correlation Between Two Skellam-Distributed Random Variables

I have two Skellam-distributed random variables, $X_1$ and $X_2$, with rate parameters $\mu_{1,+}, \mu_{1,-}$ and $\mu_{2,+}, \mu_{2,-}$ respectively. (For my case, it just so happens that their ...
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1answer
182 views

Distribution of X-Y for identical independent random variables

I have X and Y which are independent and both have an exponential distribution with density function $f(x) = e^{-x}$ if $x\gt0$ I want to find the distribution of X+Y and X-Y. Let U=X+Y, V=X-Y My ...
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1answer
20 views

bivariate distribution with joint density Integration

i have a bivariate distribution for (X, Y) with joint density: $$f(x,y) = \frac{\lambda y^2}{\sqrt2\pi}e^-(\frac{1}{2} + \lambda x)y^2$$ on x>0 and y belong to R I need to show that $f_X(x) = \frac{\...
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1answer
26 views

Find borel set such that $P(X \in A, Y \in B) \neq P(X \in A)P(Y \in B)$

I am given the pdf of the uniform distribution on the disk of radius R: $1/\pi(R^2)$ I have shown that X and Y are not independent by showing that $f_X(x)f_Y(y) \neq 1/\pi(R^2)$ However, despite ...
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0answers
23 views

Joint distribution function of max and min of independent, identically distributed random variables [duplicate]

Suppose we have independent random variables $X_1,\ldots,X_n$ with the same distribution. Let $F(\cdot)$ be the cumulative distribution of $X_i$ for all i We are interested in the cumulative ...
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1answer
12 views

Interpreting covariance in bivariate normal plot

I am a newbie in stats, and after much searching, I found this site: https://demonstrations.wolfram.com/TheBivariateNormalDistribution/ It is great, and I guess it cannot get more basic than that, ...
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1answer
18 views

different bivariate normal formulas

I do not understand the below definition for the bivariate normal... Up (formula 4-5) is defining the squared distance as a function of 2 variables $x_1$ and $x_2$. And below(formula 4-6) is the ...
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1answer
26 views

Inequality for joint distribution function

The following problem is from Rohatgi Let (X,Y) have joint density function f and joint distribution function F. Suppose that $f(x_1,y_1)f(x_2,y_2) \le f(x_1,y_2)f(x_2,y_1)$ holds for any $x_1 \le a \...
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2answers
45 views

P(X<Y) given joint density function

I am given the joint density function $f(x,y)$ for random variables X,Y with $0<x<1$, $0<y<2$ I am interested in $P(X<Y)$ My first instinct was to do the following: $$P(X<Y) = \...
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0answers
31 views

Marginal probability mass function of bivariate negative binomial

define $$P(X=x,Y=y) = {(x+y+k-1)!\over x!y!(k-1)!}p_1^xp_2^y(1-p_1-p_2)^k$$ the bivariate negative binomial distribution. I am interested in the marginal probability mass function of X. After ...
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0answers
26 views

How to integrate this function in correct way?

I got an exercise on my class about bivariat distribution. Given $f(x,y)= x+y \quad \text{for} \quad 0<x<1, 0<y<1\\ f(x,y)= 0 \quad \text{for other.}\\$ Find $P[X+Y \leq \frac{3}{2}]$. I ...
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8 views

Is there terminology around a bivariate function that is dependent on only one of its two arguments?

For example, if $f$ can be defined by $f(x, y) = g(x)$ for some $g$ and all $x$, are there any special adjectives that are applicable to $f$?
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1answer
35 views

Estimate Percentile Rank from a Bivariate Normal with know correlation coefficient

The problems goes: Suppose the scores of an exam follow Normal distribution and the correlation coefficient $\rho=0.8$ for exam1 and exam2. If in exam1 a student's score percentage rank (PR) is 90% (...
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1answer
46 views

Bivariate Gaussian Variables: Finding the distribution of the conditional probability

Exercise question (from Timo Koski's book Lecture Notes: Probability and Random Processes at KTH): Let $(X_{1},X_{2})'\in N(\pmb{\mu},\pmb{C})$, where $$\pmb{\mu}=\begin{pmatrix} 0\\ 0 \end{pmatrix}...
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1answer
34 views

Understand simplification step in deriving the conditional bivariate normal distribution

I'm reading through Ross's A First Course in Probability , and am having some difficulty understanding a step within the derivation of the conditional bivariate normal. $$f(x, y) = \frac{1}{2\pi\...
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15 views

Bivariate Copula

Identify the copula of the bivariate CDF $F(x,y) = (1-\exp(-2x))(1-\exp(-y^2))/(1-\theta*exp(-2x-y^2))$. Can anyone provide any hint? I calculated the marginals by $F(x,\infty)= 1-e^{-2x}$
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1answer
75 views

Does a joint density $ f_{X,Y} $ split uniquely into two marginal PDF's $ f_{X} (x) $ and $f_{Y} (y) $ for independent random variables?

I was doing a question and I came across the following logic: Claim: If you have $X$ and $Y$ as two random variables and $$f_{X,Y} (x,y) = \frac{1}{2\sqrt{3} \pi} e^{-\frac{1}{2}\left[\frac{...
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0answers
26 views

Bivariate Gaussian copula family is ordered

The bivariate gaussian copula is defined as $$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp⁡(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy$$ where $\Phi$ is the cumulative ...
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0answers
37 views

Asymptotic variance of maximum likelihood estimate

I am stuck at the part of the asymptotic variance of the MLE. The MLE I derived is $\hat{\theta} = \frac{\sum_i x_i y_i}{\sum_i x_i^2}$. Using iterated expectation, I can prove $\hat{\theta}$ is ...